A $\emph{convex}$ polygon is a polygon in which every interior angle is less than 180 degrees.  A $\emph{diagonal}$ of a convex polygon is a line segment that connects two non-adjacent vertices.  How many diagonals does a convex polygon with 20 sides have?
For each of the 20 vertices of the polygon, there are 17 other nonadjacent vertices that we can connect the original vertex with to form a diagonal.  However, multiplying 20 by 17 would count each diagonal twice---once for each of the diagonal's endpoints. We must divide the result by 2 to correct for this, so the answer is $(20\cdot 17)/2=\boxed{170}$.